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Mirrors > Home > MPE Home > Th. List > ellspsn6 | Structured version Visualization version GIF version |
Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
ellspsn5b.v | ⊢ 𝑉 = (Base‘𝑊) |
ellspsn5b.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
ellspsn5b.n | ⊢ 𝑁 = (LSpan‘𝑊) |
ellspsn5b.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ellspsn5b.a | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Ref | Expression |
---|---|
ellspsn6 | ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellspsn5b.a | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
2 | ellspsn5b.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
3 | ellspsn5b.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | 2, 3 | lssel 20953 | . . . 4 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
5 | 1, 4 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
6 | ellspsn5b.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑊 ∈ LMod) |
8 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
9 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
10 | ellspsn5b.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
11 | 3, 10 | lspsnss 21006 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
12 | 7, 8, 9, 11 | syl3anc 1370 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
13 | 5, 12 | jca 511 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)) |
14 | 2, 10 | lspsnid 21009 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
15 | 6, 14 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
16 | ssel 3989 | . . . 4 ⊢ ((𝑁‘{𝑋}) ⊆ 𝑈 → (𝑋 ∈ (𝑁‘{𝑋}) → 𝑋 ∈ 𝑈)) | |
17 | 15, 16 | syl5com 31 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) ⊆ 𝑈 → 𝑋 ∈ 𝑈)) |
18 | 17 | impr 454 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)) → 𝑋 ∈ 𝑈) |
19 | 13, 18 | impbida 801 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 ‘cfv 6563 Basecbs 17245 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-lmod 20877 df-lss 20948 df-lsp 20988 |
This theorem is referenced by: ellspsn5b 21011 lsmelval2 21102 dihjat1lem 41411 |
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